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Schrödinger–Maxwell systems on compact Riemannian manifolds

Csaba Farkas

1,2

1Department of Mathematics and Computer Science, Sapientia University, Târgu Mures,, Romania

2Institute of Applied Mathematics, Óbuda University, 1034 Budapest, Hungary

Received 24 March 2018, appeared 26 July 2018 Communicated by Dimitri Mugnai

Abstract. In this paper, we are focusing to the following Schrödinger–Maxwell system:

(gu+β(x)u+euφ=Ψ(λ,x)f(u) inM,

gφ+φ=qu2 inM, (S MeΨ(λ,·))

where (M,g) is a 3-dimensional compact Riemannian manifold without boundary, e,q > 0 are positive numbers, f : RR is a continuous function, βC(M) and ΨC(R+×M) are positive functions. By various variational approaches, existence of multiple solutions of the problem(S MeΨ(λ,·))is established.

Keywords: Schrödinger–Maxwell systems, critical points, compact Riemannian mani- folds.

2010 Mathematics Subject Classification: 58J05, 35A01, 35J20, 35J47, 35J61, 35R01, 58E05.

1 Introduction and statement of the main results

We are concerned with the nonlinear Schrödinger–Maxwell system (−gu+β(x)u+euφ=Ψ(λ,x)f(u) in M,

gφ+φ=qu2 in M, (S MeΨ(λ,·)) where (M,g) is a 3-dimensional compact Riemannian manifold without boundary, e,q > 0 are positive numbers, f :RRis a continuous function, β∈C(M)andΨ∈C(R+×M) are positive functions.

From physical point of view, the Schrödinger–Maxwell systems (−2m¯h2∆u+ωu+euφ= f(x,u) inR3,

∆φ=4πeu2 inR3, (1.1)

describe the statical behavior of a charged non-relativistic quantum mechanical particle inter- acting with the electromagnetic field. More precisely, the unknown terms u : R3R and

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φ : R3R are the fields associated to the particle and the electric potential, respectively, the nonlinear term f models the interaction between the particles and the coupled term φu concerns the interaction with the electric field. Note that the quantitiesm, e, ω and ¯h are the mass, charge, phase, and Planck’s constant.

In fact, system (1.1) comes from the evolutionary nonlinear Schrödinger equation by using a Lyapunov–Schmidt reduction.

The Schrödinger–Maxwell system (or its variants) has been the object of various investiga- tions in the last two decades, the existence/non-existence of positive solutions, sign-changing solutions, ground states, radial, non-radial solutions, and semi-classical states has been in- vestigated by several authors. Without sake of completeness, we recall in the sequel some important contributions to the study of system (1.1). Benci and Fortunato [7] considered the case of f(x,s) = |s|p2s with p ∈ (4, 6) by proving the existence of infinitely many radial solutions for (1.1); their main step relies on the reduction of system (1.1) to the investigation of critical points of a “one-variable” energy functional associated with (1.1).

Based on the idea of Benci and Fortunato, under various growth assumptions on f further existence/multiplicity results can be found in Ambrosetti and Ruiz [4], Azzolini [5], in [6]

Azzollini, d’Avenia, and Pomponio were concerned with the existence of a positive radial solution to system (1.1) under the effect of a general nonlinear term, in [11] the existence of a non radially symmetric solution was established when p∈ (4, 6), by means of a Pohozaev- type identity, d’Aprile and Mugnai [12,13] proved the non-existence of non-trivial solutions to system (1.1) whenever f ≡ 0 or f(x,s) = |s|p2s and p ∈ (0, 2]∪[6,∞), the same authors proved the existence of a non-trivial radial solution to (1.1), for p∈ [4.6). Other existence and multiplicity result can be found in the works of Cerami and Vaira [8], Kristály and Repovs [23], Ruiz [27], Sun, Chen, and Nieto [28], Wang and Zhou [31], and references therein.

In the last five years Schrödinger–Maxwell systems has been studied on n−dimensional compact or non-compact Riemannian manifolds(2≤n≤ 5) by Druet and Hebey [14], Farkas and Kristály [16], Hebey and Wei [19], Ghimenti and Micheletti [17,18] and Thizy [29,30]. More precisely, in the aforementioned papers various forms of the system

(−¯hm2gu+ωu+euφ= f(x,u) in M,

gφ+φ=4πeu2 in M, (1.2)

have been considered, where(M,g)is a Riemannian manifold.

The aim of this paper is threefold. First, we consider the system (S MeΨ(λ,·)) withΨ(λ,x) = λα(x), whereαis a suitable function and we assume that f is a sublinear nonlinearity (see the assumptions (f1)–(f3) below). In this case, we prove that if the parameterλ is small enough the system (S Meλ) has only the trivial solution, while if λ is large enough then the system (S MeΨ(λ,·)) has at least two solutions, see Theorem 1.1. It is natural to ask what happens between this two threshold values. In this gap interval we have no information on the number of solutions (S MeΨ(λ,·)); in the case when q→ 0 these two threshold values may be arbitrary close to each other. Similar bifurcation type result for a perturbed sublinear elliptic problem was obtained by Kristály, see [20].

Second, we consider the system (S MλΨ(λ,·)) withΨ(λ,x) =λα(x) +µ0β(x), whereαandβ are suitable functions. In order to prove a new kind of multiplicity for the system (S MλΨ(λ,·)) (i.e.e =λ), we show that certain properties of the nonlinearity, concerning the set of all global minima, can be reflected to the energy functional associated to the problem, see Theorem1.3.

Third, as a counterpart of Theorem 1.1 we will consider the system (S MeΨ(λ,·)) with Ψ(λ,x) = λ, and f here satisfies the so called Ambrosetti–Rabinowitz condition. This type

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of result is motivated by the result of Anello [3] and Ricceri [24], where the authors studied the classical Ambrosetti–Rabinowitz problem, without the assumption limt0 f(t)

t =0, i.e. the authors proved that if the nonlinearity f satisfies the so called (AR) condition and a subcritical growth condition, then if λis small enough the problem

(−∆u= λf(u) inΩ,

u=0 onΩ,

has at least two weak solutions in H01().

In the sequel we present precisely our results. As we mentioned before, we first consider a continuous function f :[0,)→Rwhich verifies the following assumptions:

(f1) f(ss)0 ass →0+; (f2) f(ss) →0 ass →∞;

(f3) F(s0)>0 for somes0>0, where F(s) =Rs

0 f(t)dt,s ≥0.

Due to the assumptions (f1)–(f3), the numbers cf =max

s>0

f(s) s and

cF =max

s>0

4F(s) 2s2+eqs4

are well-defined and positive. Now, we are in the position to state the first result of the paper.

In order to do this, first we recall the definition of the weak solutions of the problem (S Meλ): The pair (u,φ)∈ Hg1(M)×Hg1(M)is aweak solutionto the system(S Meλ)if

Z

M

(h∇gu,∇gvi+β(x)uv+euφv)dvg =

Z

MΨ(λ,x)f(u)vdvgfor allv ∈ Hg1(M), (1.3) Z

M

(h∇gφ,gψi+φψ)dvg =q Z

Mu2ψdvg for allψ∈ H1g(M). (1.4) Our first result reads as follows.

Theorem 1.1. Let (M,g) be a3-dimensional compact Riemannian manifold without boundary, and let β ≡ 1. Assume that Ψ(λ,x) = λα(x) andα∈ C(M)is a positive function. If the continuous function f :[0,∞)→Rsatisfies assumptions(f1)–(f3), then

(a) if0≤λ<cf1kαkL1, system(S MλΨ(λ,·))has only the trivial solution;

(b) for every λ≥ cF1kαkL11, system(S MλΨ(λ,·))has at least two distinct non-zero, non-negative weak solutions in H1g(M)×H1g(M).

Remark 1.2.

(a) Due to (f1), it is clear that f(0) = 0, thus we can extend continuously the function f :[0,)→Rto the wholeRby f(s) =0 fors ≤0; thus,F(s) =0 fors ≤0.

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(b) (f1) and (f2) mean that f is superlinear at the origin and sublinear at infinity, respectively.

Typical functions which fulfill hypotheses (f1)–(f3) are

f(s) =min(sr,sp), 0<r<1< p, s ≥0 or

f(s) =ln(1+s2), s≥0.

(c) By a three critical points result of Ricceri [26], one can prove that the number of solutions of the problem (S MeΨ(λ,·)) for λ > λ˜ is stable under small nonlinear perturbations g : RRof subcritical type, i.e., g(s) =o(|s|21)as|s| →∞, 2 = N2N2, N>2.

In order to obtain new kind of multiplicity result for the system (S MλΨ(λ,·)) (with the choicee= λ), instead of the assumption (f1) we require the following one:

(f4) There existsµ0>0 such that the set of all global minima of the function t7→Φµ0(t):= 1

2t2µ0F(t) has at leastm≥2 connected components.

In this case we can state the following result.

Theorem 1.3. Let (M,g)be a 3-dimensional compact Riemannian manifold without boundary. Let f : [0,∞) → R be a continuous function which satisfies (f2) and (f4), β ∈ C(M) is a positive function. Assume that Ψ(λ,x) = λα(x) +µ0β(x), whereα ∈ C(M)is a positive function. Then for every τ > kβkL1(M)inf

t Φµ0(t)there exists λτ > 0 such that for every λ ∈ (0,λτ)the problem (S MλΨ(λ,·))has at least m+1weak solutions, m of which satisfy the inequality

1 2

Z

M

|∇gu|2+β(x)u2

dvgµ0

Z

Mβ(x)F(u)dvg <τ.

Remark 1.4. Taking into account the result of Cordaro [10] and Anello [2] one can prove the following: consider the following system:

(−gu+α(x)u+λφu=α(x)f(u) +λg(x,u), in M

gφ+φ=qu2, in M

where α ∈ L(M) with ess infα > 0, f : RR is a continuous function and g : M× RR, besides being a Carathéodory function, is such that, for some p > 3(= dimM), sup|s|≤tg(·,s)∈ Lp(M)andg(·,t)∈ L(M)for allt ∈R. If the set

Gf =

t∈ R: 1 2t2

Z t

0

f(s)ds= inf

ξR

1 2ξ2

Z ξ

0

f(s)ds

hasm≥ 2 bounded connected components, then the system has at leastm+m2weak solu- tions. For the proof, one can use a truncation argument combining with the abstract critical point theory result of Anello [1, Theorem 2.1]. Note that the similar truncation method which was presented in [10] fails, due to the extra termR

Mφuu2. To overcome this difficulty, one can use the same method as in [16, Proposition 3.1 (i) & (ii)] (see also [21]).

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Note also that similar multiplicity results was obtained by Kristály and R˘adulescu in [22], for Emden–Fowler type equations.

Our abstract tool for proving the Theorem 1.3 is the following abstract theorem that we recall here (see [25]).

Theorem A. Let H be a separable and reflexive real Banach space, and let N,G : H → R be two sequentially weakly lower semi-continuous and continuously Gateaux differentiable functionals, with N coercive. Assume that the functional N +λG satisfies the Palais–Smale condition for everyλ>0 small enough and that the set of all global minima ofN has at least mconnected components in the weak topology, withm≥2. Then, for everyη>infHN, there exists λ > 0 such that for every λ ∈ (0,λ)the functional N +λG has at least m+1 critical points, mof which are inN1((−∞,η)).

Finally, as a counterpart of the Theorem 1.1 we consider the case when the continuous function f :[0,+)→Rsatisfies the following assumptions:

( ˜f1) |f(s)| ≤C(1+|s|p1), for alls∈R, wherep ∈(2, 6); ( ˜f2) there existsη>4 andτ0 >0 such that

0<ηF(s)≤s f(s), ∀|s| ≥τ0.

Theorem 1.5. Let(M,g)be a3-dimensional compact Riemannian manifold without boundary, and let β≡1. Assume thatΨ(λ,x) =λ. Let f :RRbe a continuous function, which satisfies hypotheses (f˜1), (f˜2). Then there existsλ0 such that for every0 < λ < λ0 the problem(S Meλ)has at least two weak solutions.

Our abstract tool for proving the previous theorem is the following abstract theorem that we recall here (see [24]).

Theorem B. LetEbe a reflexive real Banach space, and letΦ,Ψ: E→Rbe two continuously Gâteaux differentiable functionals such that Φis sequentially weakly lower semi-continuous and coercive. Further, assume that Ψis sequentially weakly continuous. In addition, assume that for each µ > 0, the functional Jµ := µΦ−Ψ satisfies the classical compactness Palais–

Smale condition. Then for eachρ>infEΦand each

µ> inf

uΦ1((−∞,ρ))

supvΦ1((−∞,ρ))Ψ(v)−Ψ(u) ρΦ(u) ,

the following alternative holds: either the functional Jµ has a strict global minimum which lies in Φ1((−∞,ρ)), or Jµ has at least two critical points one of which lies inΦ1((−∞,ρ)).

2 Proof of the main results

Let β∈C(M)be a positive function. For everyu∈C(M)let us denote by kuk2β =

Z

M

|∇gu|2+β(x)u2dvg.

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The Sobolev spaceH1β is defined as the completion ofC(M)with respect to the normk · kβ. Clearly, Hβ1 is a Hilbert space. Note that, since βis positive, the norm k · kβ is equivalent to the standard norm, i.e. we have that

min

1, min

M

q β(x)

kukH1

g(M)≤ kukβ ≤maxn 1,q

kβkL(M)

okukH1

g(M). (2.1) Note that H1β(M)is compactly embedded in Lp(M), p ∈ [1, 6); the Sobolev embedding con- stant will be denoted byκp.

We define the energy functional Jλ : H1g(M)×Hg1(M) → R associated with system (S Meλ), namely,

Jλ(u,φ) = 1

2kuk2β+ e 2

Z

Mφu2dvge 4q

Z

M

|∇gφ|2dvge 4q

Z

Mφ2dvg

Z

MΨ(x,λ)F(u)dvg. It is easy to see that the functionalJλ is well-defined and of classC1 on H1g(M)×H1g(M). Moreover, due to relations (1.3) and (1.4) the pair(u,φ)∈ Hg1(M)×H1g(M)is a weak solution of(S Meλ)if and only if(u,φ)is a critical point of Jλ.

Using the Lax–Milgram theorem one can see that the equation

gφ+φ=qu2, in M

has a unique solution for any fixedu. By exploring an idea of Benci and Fortunato [7], we introduce the map φu : Hg1(M) → H1g(M) by associating to every u ∈ H1g(M) the unique solution φ = φu of the Maxwell equation. Thus, one can define the “one-variable” energy functionalEλ: H1g(M)→Rassociated with system(S Meλ):

Eλ(u) = 1

2kuk2β+ e 4

Z

Mφuu2dvg− F(u), (2.2) whereF :Hg1(M)→Ris the functional defined by

F(u) =

Z

MΨ(x,λ)F(u)dvg.

By using standard variational arguments, one has that the pair(u,φ)∈ Hg1(M)×Hg1(M)is a critical point ofJλ if and only if uis a critical point of Eλ and φ= φu, see for instance [16].

Moreover, we have that Eλ0(u)(v) =

Z

M

(h∇gu,∇gvi+β(x)uv+eφuuv)dvg

Z

MΨ(x,λ)f(u)vdvg. (2.3) 2.1 Schrödinger–Maxwell systems involving sublinear nonlinearity

In this section, we setΨ(x,λ) =λα(x) +µ0β(x). Recall that Eλ(u) = 1

2kuk2β+ e 4

Z

Mφuu2dvg

Z

MΨ(x,λ)F(u)dvg.

In order to apply variational methods, we prove some elementary properties of the func- tionalEλ.

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Lemma 2.1. The energy functionalEλ is coercive, for everyλ≥0.

Proof. Indeed, due to (f2), we have that for everyε > 0 there exists δ > 0 such that |F(s)| ≤ ε|s|2, for every|s|>δ. Thus, since Ψ(x,λ)∈ L(M)we have that

F(u) =

Z

{u>δ}Ψ(x,λ)F(u)dvg+

Z

{uδ}Ψ(x,λ)F(u)dvg

εkΨ,λ)kL(M)κ22kuk2β+kΨ,λ)kL(M)VolgMmax

|s|≤δ

|F(s)|. Therefore,

Eλ(u)≥ 1

2−εκ22kΨ(·,λ)kL(M)

kuk2β−VolgM· kΨ(·,λ)kL(M)max

|s|≤δ

|F(s)|. In particular, if 0<ε<(2κ22kΨ(·,λ)kL(M))1, thenEλ(u)→askukβ∞.

Lemma 2.2. The energy functionalEλ satisfies the Palais–Smale condition for everyλ≥0.

Proof. Let{uj}j ⊂ Hg1(M)be a Palais–Smale sequence, i.e.{Eλ(uj)}j is bounded and k(Eλ)0(uj)kH1

g(M) →0

as j → ∞. Since Eλ is coercive (see Lemma 2.1), the sequence {uj}j is bounded in H1g(M). Therefore, up to a subsequence, then{uj}jconverges weakly inH1g(M)and strongly inLp(M), p ∈(2, 2), to an elementu∈ H1g(M).

First we claim that for allu,v∈ H1g(M)we have that Z

M

(uφu−vφv) (u−v)dvg ≥0. (2.4) This inequality is equivalent with the following one:

Z

Mφuu2dvg+

Z

Mφvv2dvg

Z

M

(φuuv+φvuv)dvg. On the other hand, using the Cauchy–Schwarz inequality, we have, that

Z

M

(φuuv+φvuv)dvgZ

Mφuu2dvg

1/2Z

Mφuv2dvg 1/2

+ Z

Mφvu2dvg

1/2Z

Mφvv2dvg 1/2

= 1 q

Z

M

(∇gφugφv+φuφv)dvg

1/2

(kφukH1

g(M)+kφvkH1 g(M))

1

qkφuk1/2

Hg1(M)kφvk1/2

H1g(M)

kφukH1

g(M)+kφvkH1 g(M)

.

Taking into account the following algebraic inequality(xy)1/2(x+y)≤(x2+y2),(∀)x,y≥0, we have that

kφuk1/2H1

g(M)kφvk1/2H1

g(M)

kφukH1

g(M)+kφvkH1 g(M)

≤ kφuk2H1

g(M)+kφvk2H1 g(M).

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Therefore, Z

M

(φuuv+φvuv)dvg1 q

kφuk2H1

g(M)+kφvk2H1

g(M)

=

Z

Mφuu2dvg+

Z

Mφvv2dvg, which proves the claim.

Now, using inequality (2.4) one has Z

M

|∇guj− ∇gu|2dvg+

Z

Mβ(x) uj−u2

dvg

≤(Eλ)0(uj)(uj−u) + (Eλ)0(u)(u−uj) +

Z

MΨ(x,λ)[f(uj(x))− f(u(x))](uj−u)dvg. Sincek(Eλ)0(uj)kH1

g(M) → 0, anduj *uin Hg1(M), the first two terms at the right hand side tend to 0. Letp∈ (2, 2).

By the assumptions on f, for every ε>0 there exists a constantCε >0 such that

|f(s)| ≤ε|s|+Cε|s|p1,

for everys∈R. The latter relation, Hölder inequality and the fact thatuj →uinLp(M)imply

that

Z

MΨ(x,λ)[f(uj)− f(u)](uj−u)dvg

→0, asj→∞. Therefore,kuj−uk2

H1g(M)→0 as j→∞, which proves our claim.

Before we prove Theorem1.1, we prove the following lemma.

Lemma 2.3. Let f : [0,+) → R be a continuous function satisfying the assumptions (f1)–(f3) . Then

cf :=max

s>0

f(s)

s >cF :=max

s>0

4F(s) 2s2+eqs4. Proof. Lets0>0 be a maximum point for the functions7→ 4F(s)

2s2+eqs4, therefore cF = 4F(s0)

2s20+eqs40 = f(s0)

s0+eqs30f(s0) s0

≤cf. Now we assume thatcf =cF:=θ. Let

es0:=inf

s >0 :θ= 4F(s) 2s2+eqs4

.

Note thates0>0. Fix t0 ∈(0,es0), in particular 4F(t0)< θ(2t30+eqt40). On the other hand, from the definition ofcf, one has f(t)≤ θ(s+eqs3). Therefore

0=4F(es0)−θ(2es0+eqes40) =4F(t0)−θ(2t20+eqt40)+4 Z es0

t0

f(t)−θ(s+eqs3)ds<0, which is a contradiction, thuscf > cF.

Now we are in the position to prove Theorem1.1.

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Proof of Theorem1.1. First recall that, in this case, β(x) ≡ 1 and Ψ(λ,x) = λα(x), and α ∈ C(M)is a positive function.

(a) Letλ≥0. If we choosev=uin (1.3) we obtain that Z

M

|∇gu|2+u2+eφuu2

dvg =λ Z

Mα(x)f(u)udvg.

As we already mentioned, due to the assumptions (f1)–(f3), the number cf = maxs>0 f(s) s is well-defined and positive. Thus, sincekφuk2H1

g(M)= qR

Mφuu2dvg≥0, we have that kuk2H1

g(M) ≤ kuk2H1

g(M)+e Z

Mφuu2dvgλcfkαkL(M) Z

Mu2dvgλcfkαkL(M)kuk2H1

g(M). Therefore, if λ < cf1kαkL1(M), then the last inequality gives u = 0. By the Maxwell’s equation we also have thatφ=0, which concludes the proof of (a).

(b) By using assumptions(f1)and(f2), one has

Hlim(u)→0

F(u)

H(u) = lim

H(u)→

F(u) H(u) =0, where H(u) = 12kuk2β+ 4eR

Mφuu2dvg. Since α ∈ C(M)+\ {0}, on account of (f3), one can guarantee the existence of a suitable truncation function uT ∈ H1g(M)\ {0} such that F(uT)>0. Therefore, we may define

λ0= inf

uH1g(M)\{0} F(u)>0

H(u) F(u).

The above limits imply that 0<λ0<∞. Since H1g(M)contains the positive constant functions on M, we have

λ0 = inf

uH1g(M)\{0} F(u)>0

H(u)

F(u) ≤max

s>0

2s2+eqs4 4F(s)kαkL1(M)

=cF1kαk1

L1(M).

For every λ > λ0, the functional Eλ is bounded from below, coercive and satisfies the Palais–Smale condition (see Lemma 2.1, Lemma 2.2). If we fix λ > λ0 one can choose a functionw∈ H1g(M)such thatF(w)>0 and

λ> H(w) F(w) ≥λ0. In particular,

c1:= inf

H1g(M)Eλ ≤ Eλ(w) =H(w)−λF(w)<0.

The latter inequality proves that the global minimumu1λ ∈ H1g(M)of Eλ on Hg1(M)has nega- tive energy level.

In particular,(u1λ,φu1 λ

)∈ Hg1(M)×H1g(M)is a nontrivial weak solution to(S Meλ).

Letν ∈ (2, 6)be fixed. By assumptions, for anyε > 0 there exists a constantCε > 0 such that

0≤ |f(s)| ≤ ε kαkL(M)

|s|+Cε|s|ν1 for alls∈R.

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Thus

0≤ |F(u)| ≤

Z

M

α(x)|F(u(x))|dvg

Z

Mα(x) ε 2kαkL(M)

u2(x) + Cε

ν |u(x)|ν

! dvg

ε 2kuk2H1

g(M)+ Cε

ν kαkL(M)eκννkukνH1

g(M),

whereeκν is the embedding constant in the compact embeddingHg1(M),→Lν(M), ν∈ [1, 6). Therefore,

Eλ(u)≥ 1

2(1−λε)kuk2H1

g(M)λCε

ν kαkL(M)eκννkukνH1 g(M). Bearing in mind thatν>2, for enough small ρ>0 andε< λ1we have that

kukHinf1

g(M)=ρ

Eλ(u)≥ 1

2(1−ελ)ρλCε

ν kαkL(M)eκννρ

ν 2 >0.

A standard mountain pass argument (see, for instance, Willem [32]) implies the existence of a critical pointu2λ ∈ H1g(M)forEλwith positive energy level. Thus(u2λ,φu2

λ

)∈ Hg1(M)×H1g(M) is also a nontrivial weak solution to(S Meλ). Clearly,u1λ 6= u2λ.

It is also clear that the functionq 7→ maxs>0 4F(s)

2s2+eqs4 is non-increasing. Let a > 1 be a real number. Now, consider the following function

f(s) =





0, 0≤s <1, s+g(s), 1≤s <a, a+g(a), s ≥a,

whereg:[1,+)→Ris a continuous function with the following properties (g1) g(1) =−1;

(g2) the functions7→ g(ss) is non-decreasing on[1,+); (g3) lim

sg(s)<∞.

In this case the

F(s) =









0, 0≤s<1,

s2

2 +G(s)−1

2, 1≤s<a,

(a+g(a))s− a

2

2 +G(a)−ag(a)−1

2, s≥ a, whereG(s) =Rs

1 g(t)dt. It is also clear that f satisfies the assumptions (f1)–(f3).

Thus, a simple calculation shows that

cf = a+g(a)

a .

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We also claim that

bcF =lim

q0cF = (a+g(a))2

a2+2ag(a)−2G(a) +1. Indeed,

bcF=max

s>0

2F(s) s2 .

It is clear that it is enough to show that the maximum of the function 2Fs(2s) is achieved on the intervals≥ a, i.e.,

sg(s)−2G(s)>−1, s>1.

Now, using a result of [9, page 42, equation (4.3)] (see also [15, Theorem 1.3]), we have that the function Gs2(s1)

2

is increasing, thus

sg(s)−2G(s)≥ g(s)

s ≥ −1, s≥ −1, which proves our claim.

One can see from the assumptions ong, that the values cf and bcF may be arbitrary close to each other. Indeed, when

alimcf = lim

abcF =1.

Therefore, ifα≡1 then the threshold values arecf1 andcF1(which are constructed indepen- dently), i.e., if λ ∈ (0,cf1) we have just the trivial solution, while ifλ ∈ (cF1,+) we have at least two solutions. λ lying in the gap-interval [cf1,cF1] we have no information on the number of solutions for(S Meλ).

Taking into account the above example we see that if the “impact” of the Maxwell equation is small (q→0), then the valuescf andcFmay be arbitrary close to each other.

Remark 2.4. Typical examples for functiongcan be:

(a) g(s) =−1. In this casecf = aa1 and bcF= aa+11. (b) g(s) = 1s −2. In this casecf = (a1)2

a2 and bcF= (a1)4

a2(a22 lna1).

Proof of Theorem1.3. We follow the idea presented in [22]. First, we claim that the set of all global minima of the functionalN : H1g(M)→R,

N(u) = 1

2kuk2βµ0 Z

Mβ(x)F(u)dvg

has at least m connected components in the weak topology on Hg1(M). Indeed, for every u∈Hβ1(M)one has

N(u) = 1

2kuk2βµ0 Z

M

β(x)F(u)dvg

= 1 2

Z

M

|∇gu|2dvg+

Z

Mβ(x)Φµ0(u)dvg

≥ kβkL1(M)inf

t Φµ0(t).

(12)

Moreover, if we consider u = t˜ for a.e. x ∈ M, where ˜t ∈ R is the minimum point of the functiont 7→Φµ0(t), then we have equality in the previous estimate. Thus,

inf

uH1β(M)

N(u) =kβkL1(M)inf

t Φµ0(t).

On the other hand, if u ∈ H1g(M) is not a constant function, then |∇gu|2 > 0 on a positive measure set in M, i.e.

N(u)>kβkL1(M)inf

t Φµ0(t).

Consequently, there is a one-to-one correspondence between the sets Min(N) =

(

u∈ Hg1(M): N(u) = inf

uHg1(M)

N(u) )

and

Min Φµ0

=

t∈R: Φµ0(t) = inf

tRΦµ0(t)

.

Let ξ be the function that associates to every t ∈ R the equivalence class of those functions which are a.e. equal to t on the whole M. Then ξ : Min(N) → Min Φµ0

is actually a homeomorphism, where Min(N) is considered with the relativization of the weak topology on Hg1(M). On account of (f4), the set Min Φµ0

has at least m ≥ 2 connected components.

Therefore, the same is true for the set Min(N), which proves the claim.

Now, we are in the position to apply TheoremAwith H= H1g(M), N and G = 1

4 Z

Mφuu2dvg

Z

Mα(x)F(u)dvg.

Now, we prove that the functionalG is sequentially weakly lower semicontinuous. To see this, it is enough to prove that the map

Hβ1(M)3u7→

Z

Mφuu2dvg

is convex. To prove this, let us fixu,v ∈ H1β(M)andt,s ≥0 such thatt+s=1. Then we have that

A(φtu+sv):=−gφtu+sv+φtu+sv =q(tu+sv)2

≤q(tu2+sv2)

=t(qu2) +s(qv2)

=t(−gφu+φu) +s(−gφv+φv)

=A(tφu+sφv). Then, using a comparison principle it follows that

φtu+sv≤tφu+sφv.

Then, multiplying the equations−gφu+φu= qu2byφv and−gφv+φv = qv2 byφu, after integration, we obtain that

Z

M

(∇gφugφv+φuφv)dvg =q Z

Mu2φvdvg =q Z

Mv2φudvg. (2.5)

(13)

Thus, combining the above outcomes we have Z

Mφtu+sv(tu+sv)2dvg

Z

M

(tφu+sφv) tu2+sv2 dvg

= t2 Z

Mφuu2dvg+ts Z

M

φuv2+φvu2

dvg+s2 Z

Mφvv2dvg (2.5)

= t

2

q Z

M

|∇gφu|2dvg+

Z

Mφu2dvg

+ 2ts q

Z

M

(∇gφugφv+φuφv)dvg

+ s

2

q Z

M

|∇gφv|2dvg+

Z

Mφ2vdvg

= 1 q

Z

M

(t∇gφu+s∇gφv)2dvg+ 1 q

Z

M

(tφu+sφv)2dvg

≤ t Z

Mφuu2dvg+s Z

Mφvv2dvg,

which gives the required inequality, therefore it follows the required convexity. Almost the same way as in Lemma2.2we can prove that N +λG satisfies the Palais–Smale condition for everyλ >0 small enough. Therefore, the functionals N andG satisfies all the hypotheses of Theorem A. Therefore for every τ > max

0,kβk1inftΦµ0(t) there exists λτ > 0 such that for every λ ∈ (0,λτ)the problem (S Mλλ) has at leastm+1 solutions. We know in addition that m elements among the solutions belong to the set Nν01((−∞,τ)), which proves that m solutions satisfy the inequality

1 2

Z

M

|∇gu|2+β(x)u2

dvgµ0 Z

Mβ(x)F(u)dvg< τ.

Remark 2.5.

(a) Note that (f4) implies that the functiont7→Φµ0(t)has at leastm−1 local maxima. Thus, the functiont 7→µ0f(t)has at least 2m−1 fixed points. In particular, if for someλ>0

Ψ(x,λ) =µ0β(x), for everyx ∈ M, then the problem(S Mλλ)has at least 2m−1≥3 constant solutions.

(b) Using the abstract TheoremA, one can guarantee thatτ>max

0,kβkL1(M)inftΦµ0(t) It is clear that the assumption(f2) holds if there existν ∈(0, 1)andc>0 such that

|f(t)| ≤c|t|ν, for everyt∈ R.

In this case, mweak solutions of the problem satisfy the inequality 1

2 Z

M

|∇gu|2+β(x)u2

dvgµ0 Z

Mβ(x)F(u)dvg <τ.

Now, it is clear that

|F(t)| ≤ c

ν+1|t|ν+1, for everyt∈ R.

Using a Hölder inequality Z

Mβ(x)|u|ν+1dvg≤ kβk12ν

L1(M)kukν+1

H1β(M).

Hivatkozások

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